Problem: $\dfrac{ 10b - 4c }{ 3 } = \dfrac{ 3b - 5d }{ 7 }$ Solve for $b$.
Explanation: Multiply both sides by the left denominator. $\dfrac{ 10b - 4c }{ {3} } = \dfrac{ 3b - 5d }{ 7 }$ ${3} \cdot \dfrac{ 10b - 4c }{ {3} } = {3} \cdot \dfrac{ 3b - 5d }{ 7 }$ $10b - 4c = {3} \cdot \dfrac { 3b - 5d }{ 7 }$ Multiply both sides by the right denominator. $10b - 4c = 3 \cdot \dfrac{ 3b - 5d }{ {7} }$ ${7} \cdot \left( 10b - 4c \right) = {7} \cdot 3 \cdot \dfrac{ 3b - 5d }{ {7} }$ ${7} \cdot \left( 10b - 4c \right) = 3 \cdot \left( 3b - 5d \right)$ Distribute both sides ${7} \cdot \left( 10b - 4c \right) = {3} \cdot \left( 3b - 5d \right)$ ${70}b - {28}c = {9}b - {15}d$ Combine $b$ terms on the left. ${70b} - 28c = {9b} - 15d$ ${61b} - 28c = -15d$ Move the $c$ term to the right. $61b - {28c} = -15d$ $61b = -15d + {28c}$ Isolate $b$ by dividing both sides by its coefficient. ${61}b = -15d + 28c$ $b = \dfrac{ -15d + 28c }{ {61} }$